SRC/clahr2.f(3)

NAME
SRC/clahr2.f

SYNOPSIS

  Functions/Subroutines
subroutine clahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Function/Subroutine Documentation

  subroutine clahr2 (integer n, integer k, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( nb ) tau, complex, dimension( ldt, nb ) t, integer ldt, complex, dimension( ldy, nb ) y, integer ldy)

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

    !>
    !> CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
    !> matrix A so that elements below the k-th subdiagonal are zero. The
    !> reduction is performed by an unitary similarity transformation
    !> Q**H * A * Q. The routine returns the matrices V and T which determine
    !> Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.
    !>
    !> This is an auxiliary routine called by CGEHRD.
    !>

Parameters
    N

    !>          N is INTEGER
    !>          The order of the matrix A.
    !>

    K

    !>          K is INTEGER
    !>          The offset for the reduction. Elements below the k-th
    !>          subdiagonal in the first NB columns are reduced to zero.
    !>          K < N.
    !>

    NB

    !>          NB is INTEGER
    !>          The number of columns to be reduced.
    !>

    A

    !>          A is COMPLEX array, dimension (LDA,N-K+1)
    !>          On entry, the n-by-(n-k+1) general matrix A.
    !>          On exit, the elements on and above the k-th subdiagonal in
    !>          the first NB columns are overwritten with the corresponding
    !>          elements of the reduced matrix; the elements below the k-th
    !>          subdiagonal, with the array TAU, represent the matrix Q as a
    !>          product of elementary reflectors. The other columns of A are
    !>          unchanged. See Further Details.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    TAU

    !>          TAU is COMPLEX array, dimension (NB)
    !>          The scalar factors of the elementary reflectors. See Further
    !>          Details.
    !>

    T

    !>          T is COMPLEX array, dimension (LDT,NB)
    !>          The upper triangular matrix T.
    !>

    LDT

    !>          LDT is INTEGER
    !>          The leading dimension of the array T.  LDT >= NB.
    !>

    Y

    !>          Y is COMPLEX array, dimension (LDY,NB)
    !>          The n-by-nb matrix Y.
    !>

    LDY

    !>          LDY is INTEGER
    !>          The leading dimension of the array Y. LDY >= N.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

    !>
    !>  The matrix Q is represented as a product of nb elementary reflectors
    !>
    !>     Q = H(1) H(2) . . . H(nb).
    !>
    !>  Each H(i) has the form
    !>
    !>     H(i) = I - tau * v * v**H
    !>
    !>  where tau is a complex scalar, and v is a complex vector with
    !>  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
    !>  A(i+k+1:n,i), and tau in TAU(i).
    !>
    !>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
    !>  V which is needed, with T and Y, to apply the transformation to the
    !>  unreduced part of the matrix, using an update of the form:
    !>  A := (I - V*T*V**H) * (A - Y*V**H).
    !>
    !>  The contents of A on exit are illustrated by the following example
    !>  with n = 7, k = 3 and nb = 2:
    !>
    !>     ( a   a   a   a   a )
    !>     ( a   a   a   a   a )
    !>     ( a   a   a   a   a )
    !>     ( h   h   a   a   a )
    !>     ( v1  h   a   a   a )
    !>     ( v1  v2  a   a   a )
    !>     ( v1  v2  a   a   a )
    !>
    !>  where a denotes an element of the original matrix A, h denotes a
    !>  modified element of the upper Hessenberg matrix H, and vi denotes an
    !>  element of the vector defining H(i).
    !>
    !>  This subroutine is a slight modification of LAPACK-3.0's CLAHRD
    !>  incorporating improvements proposed by Quintana-Orti and Van de
    !>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
    !>  returned by the original LAPACK-3.0's CLAHRD routine. (This
    !>  subroutine is not backward compatible with LAPACK-3.0's CLAHRD.)
    !>

References:
    Gregorio Quintana-Orti and Robert van de Geijn, "Improving theperformance of reduction to Hessenberg form," ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

Definition at line 180 of file clahr2.f.

Author

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